"Cosine" approximation
Periodic and preperiodic points of the Mandelbrot set can be found by means of
polynomials g_{n}(c) = f_{c}^{on}(0) . A few of
these polynomials are plotted below near the preperiodic point
m_{o} = 2 (the boundary crisis point).
As we know in the vicinity of any preperiodic point the Mandelbrot set is
selfsimilar. Let at the crisis point c = 2 +
e /r_{n} .
H.Hurwitz et al. have shown that for
e /r_{n}
sufficiently small
g_{n}(2 +
e /r_{n})
> 2 cos(e^{ 1/2})
where r_{n} = 4^{n}/6 and
n > 1 . A few of the polynomials g_{n}(2 +
e /r_{n})
are plotted as a function of e in Fig.2.
As n increases (5, 6, 7...) the polynomials approach the
universal function 2 cos(e ^{1/2}) .
The range  in e  of good approximation
rapidly increases with increasing n.
The superstable values of c (near 2 for n sufficiently
large) are associated with the roots of
cos(e ^{1/2}). We denote the real roots
of g_{n} by c_{n,j} where j = 1
corresponds to the root closest to c = 2 with the kneading sequence
CLR^{n2}, j = 2, to the next closest with kneading
sequence CLR^{n3}L and so on
c_{n,j} = 2 + 6p
^{2}(j  1/2)^{2} / 4^{n} .
For n = 25 the cosine approximation yields more then 10,000
real roots closest to c = 2 with an error of less then 1%.
One can get an approximate formula for locations of preperiodic points too.
The generalisation of universal scaling constants for these Mset copies
are [1]
d_{n,j} =
16 ^{n}/[6p ^{2}
(2j  1)^{2}] ,
a_{n,j} =
4 ^{n}/[2p (2j  1)] .
[1] H.Hurwitz, M.Frame, D.Peak "Scaling symmetries in nonlinear
dynamics. A view from parameter space" Physica D 81 (1995) 23.
Tip of the Mandelbrot set
The Mset near the preperiodic point
m_{o} = 2 is selfsimilar under scaling by a factor of
l = 4 , therefore every next
CLR^{n} (n = 1,2...) orbit is 4 times closer to
m_{o} . Corresponding miniature Mset shrinks by a factor of
l^{2} and eventually disappear.
They converge to m_{o} with symbolic dynamics [CL]R
where [CL] is preperiodic "tail" of the point and R
is its periodic part. The point is the tip of the Mandelbrot set antenna.

Periodn orbit CLR^{n} for large n gets near
unstable period1 orbit, leaves it and eventually returns into the z = 0
point. Period4 and period5 orbits CLR^{2} and
CLR^{3} are shown here. Preperiodic point with the pattern
[CLR^{3}]L and c = 1.97393 lies between these two
orbits.

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updated 29 October 2002